HYACINTHOS 28520

 

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, A'B'C' the pedal triangle of O , P a point and PaPbPc the pedal triangle of P.

 

Denote:

 

(N), (Na), (Nb), (Nc) = the NPCs of ABC, PBC, PCA, PAB, resp.

 

The line A'N intersects again (Na) at A"

The line A'Na intersects again (N) at A*

 

The line B'N intersects again (Nb) at B"

The line B'Nb intersects again (N) at B*

 

The line C'N intersects again (Nc) at C"

The line C'Nc intersects again (N) at C*  

 

Ma, Mb, Mc = the midpoints of A"A*, B"B*, C"C*, resp.

 

For P = I:

PaPbPc, MaMbMc are orthologic.

Orthologic centers?

 

Locus?

 

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[Ercole Suppa]

 

 

*** For P = I 

 

-- the orthologic center of PaPbPc and MaMbMc is the point:

 

Q1=W(I) = a (a+b-c) (a-b+c) (2 a^3-a^2 b-2 a b^2+b^3-a^2 c+2 a b c-2 b^2 c-2 a c^2-2 b c^2+c^3) (a^4-a^3 b+a b^3-b^4-a^3 c+3 a^2 b c-a b^2 c-a b c^2+2 b^2 c^2+a c^3-c^4) : : (barys)

 

= lies on these lines : {1319,5901} 

 

= (6-8-13) search numbers [0.556667262208870902, 0.412719855649028856, 3.09801199928480332]

 

 

-- the orthologic center of MaMbMc and PaPbPc is the point Q2=X(3754)

 

 

*** locus ?

 

 

Best regards

Ercole Suppa